Unit 8, Lesson 13
Completing the Square (Part 2)
Algebra 1
13.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-REI.A
Understand solving equations as a process of reasoning and explain the reasoning.
Building Toward
Materials
NoneActivity Narrative
This Math Talk focuses on operations with fractions to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve quadratic equations involving fractions.
To do the operations, students need to look for and make use of structure (MP7).
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneSolve each equation mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Advances: Speaking, Representing
13.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Scientific calculatorsActivity Narrative
In this activity, students analyze worked examples of equations solved by completing the square. The work allows them to further develop their understanding of the method when used with rational numbers and to notice common errors. Identifying errors in worked examples is an opportunity to attend to precision (MP6) and to analyze and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 3–4.
Display the four equations in the Task Statement for all to see:
Assign one equation (or more, depending on time constraints) for each group to solve by completing the square. Once group members agree on the solutions, ask them to look for errors in the worked solution for the same equation.
Engagement: Develop Effort and Persistence. Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2 students to share how they identified the error in the first worked solution. Record their thinking on a display for all to see, and keep the work visible as students continue.
Supports accessibility for: Organization; Attention
Supports accessibility for: Organization; Attention
13.3
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
In this activity, students encounter equations that contain rational numbers. They learn that, even though the numbers are not as straightforward as in equations they have previously seen, they can still solve the equations by completing the square. The less friendly numbers and the more elaborate equations make the computations more challenging, so students need to attend to precision (MP6) and mind the steps for rearranging parts of the equations, as well as the signs of the numbers—especially negative numbers.
As students work, they may need to be reminded to rewrite the equation in a different form, isolate a term to make it easier to complete the square, apply the zero product property only when a product is equal to zero, and so on. Make note of common errors or hurdles and address them during whole-class discussion.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Arrange students in groups of 2.
Display the equation , and ask students how they might solve it without graphing. If it does not come up, remind students that the zero product property does not work in this form and it is not set up for completing the square. Ask, “How could we rewrite this equation so that it is set up to complete the square?” (It should be in standard form, so the left side of the equation needs to be expanded.)
If there is not enough time for students to answer all the questions, consider asking students to look at the list of equations and to choose one they think looks easier (or more familiar) and one that looks harder (or less familiar).
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language students use to generalize the process of completing the square. Display words and phrases, such as “coefficient of the linear term,” “perfect square,” “standard form,” “numerator,” and “denominator.”
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the equations to solve. Encourage students to select the problems ahead of time, and persist with the problems they have chosen for the duration of the task (not reselecting if a problem becomes challenging for them).
Supports accessibility for: Organization, Social-Emotional Functioning
Supports accessibility for: Organization, Social-Emotional Functioning
Lesson Synthesis
To help students generalize the process of solving quadratic equations by completing the square, ask students,
- “How was the process of completing the square in this lesson different from that in an earlier lesson?” (The numbers in the equations are not as simple, so the calculations may be more involved and time consuming. In some cases, more steps are needed to complete the square, and the process may be more prone to error.)
- “How was the process of completing the square in this lesson like that in the previous lesson?” (The steps are essentially the same: turn one side into a perfect-square expression, write it as a squared factor, and reason about the value of the factor.)
- “What are some errors you have seen or have made when the solving process involves multiple steps, especially when the numbers are tricky to work with?” (Some common errors:
- Only positive solutions are shown. Negative solutions are neglected.
- Different amounts are added to the two sides of the equal sign.
- The wrong amount is added so the quadratic expression is not a perfect square.
- Incorrect calculations or missed steps.)
- “When might we want to choose to solve quadratic equations by rewriting them in factored form instead of by completing the square? When might it be preferable to complete the square?” (When we can easily see the factors in a quadratic equation, it makes sense to solve the equation by putting it in factored form. When it is not obvious what the factors are, or when the equation involves nonintegers, completing the square may be preferable.)
Emphasize that completing the square is a handy technique that can be used to solve any quadratic equation, but depending on the numbers in the equation, it may or may not be an efficient method.
Student Lesson Summary
Completing the square can be a useful method for solving quadratic equations in cases in which it is not easy to rewrite an expression in factored form. For example, let’s solve this equation:
First, we’ll add to each side to make things easier on ourselves.
To complete the square, take of the coefficient of the linear term, 5, which is , and square it, which is . Add this to each side:
Notice that is equal to 25, and rewrite it:
Since the left side is now a perfect square, let’s rewrite it:
For this equation to be true, one of these equations must true:
To finish up, we can subtract from each side of the equal sign in each equation.
It takes some practice to become proficient at completing the square, but it makes it possible to solve many more equations than we could by methods we learned previously.